The purpose of this project is to conduct research in mathematical statistics, probability, and applied mathematics in order to develop new statistical methodology applicable to biomedical sciences. We developed methods for adjusting for multiple comparisons in cohort studies that examine many different types of cancers, and implemented methods for adjusting for greater than Poisson variation in the comparison of observed numbers of cancers to expected numbers of cancers in such studies. We continued to investigate methods for analyzing geographic variation in disease rates from small geographic units, such as U.S. counties, implementing methods for adjusting for important covariates, such as latitude or ultraviolet radiation indices in examining skin melanoma rates. We continued research into methods for designing epidemiologic studies with maximum power, deriving sample size formulas for strata-matched case-control studies when examining multiple risk factors and for stratified studies examining the relative risk. We derived a test for genetic equilibrium in stratified data, and continued investigations into optimal analyses of HLA-type genetic systems. We are investigating optimal methods for estimating the attributable risk, or etiologic fraction, and calculating confidence intervals which correct for the negative bias in most current methods. A simulation study is being performed to investigate the performance of a newly derived bias-corrected estimator of attributable risk. We continue investigations into methods for examining birth cohort and calendar period trends in disease rates, including a simulation study of the performance of novel nonparametric methods we derived to test for changes in the slope of birth cohort trends. A computer program is being developed to implement parametric methods for examining birth cohort and calendar period patterns of risk in age-period-cohort models. We continue developing methods for examining mutational spectra in a defined DNA sequence, including a test for whether the number of tandem mutations is greater than expected by chance in a specific gene.